The seminars will take place at the Katholieke Universiteit
Leuven, Auditorium 200N,
Celestijnenlaan, 200, 3001 Leuven (Heverlee), and at CESAME,
Auditoire Euler, av. Georges Lemaître, 4, 1348 Louvain-la-Neuve. All seminars will be given on Friday
afternoons from 14h00 till 17h30 with a 30 minutes break around 15h30.
Students registering for
this course may request an evaluation, which can be arranged in the form of a
short report or additional presentation on one of the topics addressed in these
seminars.
How to reach Auditorium 200N in
Leuven
Click here for directions
and a map
(50m from
the Dept. of Computer Science, building 200A)
How to reach Auditoire Barb06 in Louvain-la-Neuve
Click here for directions
and a map of
Louvain-la-Neuve and a more detailed map around
Auditoire SUD01 (we recommend Parking 11)
For organizational reasons, we require that you register by contacting deboeck@auto.ucl.ac.be
or via the electronic form
on the web. The admission is free for doctoral students and participants from
Belgian academic institutions.
LIST OF SEMINARS,
DATES AND LOCATIONS
April 15, 2005, KUL, Auditorium
200N
14:00-17:30 Krylov methods : an introduction slides
Paul Van Dooren
(Université Catholique de Louvain, Belgium)
Abstract :
This introduction derives the general ideas
of Krylov based methods starting from two matrix decompositions: one based on
orthogonal similarity transformations and one based on arbitrary similarity
transformations. From these matrix decompositions we obtain the basic
recurrence relations of the Arnoldi and Lanczos algorithms and discuss possible
breakdowns. We then show how these lead to different classes of solvers for
linear systems of equations and to projection methods for the computation of
the spectrum of a matrix. This talk develops the basic background for the more
specialized lectures in this series.
April
22, 2005, UCL, Auditoire BARB06, Hall Ste Barbe
14:00-17:30 Krylov subspace
methods : practical variants slides
Henk
van der Vorst (Utrecht University, the Netherlands)
Abstract :
We will present an overview of a number of
related iterative methods for the solution of linear systems of equations. These
methods are so-called Krylov projection type methods and they include popular
methods as Conjugate Gradients, Bi-Conjugate Gradients, QMR, LSQR and GMRES. We
will discuss some useful variants of these methods, in particular Bi-CGSTAB and
GMRESR. Rounding errors may have
serious impact on the results of iterative methods and we will discuss these
effects and we will see how these effects can be repaired.
Iterative methods are often used in
combination with so-called preconditioning operators (approximations for the
inverses of the operator of the system to be solved) in order to speed up
convergence. We will give a short overview of these techniques, details will be
given in other lectures in this series.
April 29,
2005, KUL, Auditorium 200N
14:00-17:30 Iterative methods for large linear
eigenvalue problems slides
Henk van der
Vorst (Utrecht University, the Netherlands)
Abstract :
Suppose we want to compute one or more
eigenvalues and their corresponding eigenvectors of the n x n matrix A. Several
iterative methods are available: Jacobi's method, the power method, the method
of Lanczos, Arnoldi's method, and Davidson's method. We will discuss how these
methods work and how they may be used successfully. Most attention will be
given to the methods of Lanczos and Arnoldi. It is quite easy to use these
methods, but it is not so easy to select the right information from the
produced output. A variant of the Arnoldi method is known as Davidson's method
and this method has been reported as being quite successful, most notably in
connection with certain symmetric problems in computational chemistry. The
success of the method seems to depend quite heavily on (strong) diagonal
dominance of A. However, as we will
show, Davidson's method has an interesting connection with an old and almost
forgotten method of Jacobi. This leads to another view on the method of
Davidson, that may help us to explain the behaviour of the method, and that
leads to newpowerful algorithms for non-diagonally dominant and unsymmetric
matrices as well, the so-called Jacobi-Davidson methods. From these algorithms,
we can obtain essentially different approximations for eigenvalues and these
can be used to our advantage.
May 13, 2005, UCL, Auditoire SUD01, Croix-SUD (new location !)
14:00-17:30 Preconditioning
of iterative solvers for PDE’s slides
Andy
Wathen (Oxford University, U.K.)
Abstract :
This seminar will describe preconditioning approaches
for linear systems deriving from finite difference and finite element
approximation of partial differential equations. We will cover self-adjoint
problems which give rise to symmetric and positive definite matrices,
saddle-point systems which yield symmetric indefinite matrices and
non-self-adjoint problems where non-symmetric matrices result. Most of the
examples relate to problems of fluid flow, though the coverage in the lectures
should be such that familiarity with PDE’s in general and the incompressible
Navier-Stokes equations in particular will not be a prerequisite. The approach
will be rather algebraic and relevant PDE theory will only be introduced as and
when required.
May
20, 2005, KUL, Auditorium 200N
14:00-17:30 Algebraic multigrid
and multilevel methods: a general introduction slides
Yvan Notay
(Université Libre de Bruxelles, Belgium)
Abstract :
Multigrid and multilevel methods, alone or as
preconditioner for (Krylov based) iterative methods, are increasingly popular
as applications grow in size. They indeed allow to solve discrete PDEs with
memory and CPU requirements that remain proportional to the number of unknowns.
To face increasingly complex situations, there is in particular a high interest
in "algebraic" methods, that can be used robustly in a "black
box" fashion, independently of the underlying geometry or discretization.
In this lecture, we first introduce the
different type of multigrid & multilevel methods, illustrating their basic
principles and ideas with simple "geometric" examples. We next show
how each of these methods is easily described algebraically using a few key
ingredients. We then browse available theoretical results to identify the
necessary and sufficient conditions for fast convergence. In passing, this
allows to display some potential weaknesses of "geometric"
algorithms. On this basis, we discuss how the different key ingredients may be
defined by algebraic manipulations, reviewing several approaches that have been
proposed in the literature. We conclude with a few illustrations.
